More or less the A-tool bites/grips around the cylinder of the lock on the door. You then pry the cylinder out of the door. Once out you actuate the bolt with a bolt tool. Very similar in theory of operation as a K-tool.

The firefighter assigned forcible entry has this tool available to him, it usually lays on the step so that he can grab it once he's out if the rig. Normally, they only bring it if they think they'll need it. We will use it to pull locks on commercial aluminum doors with Adams L-R mortise locks, and in apartment buildings to pull the mortised lock and then flip the tool over and drive the mechanism fastened to the inside of the door off with the chisel end. Beyond these two uses there is not much other we use it for. We use it in an attempt to not damage the door itself. As for tubular deadbolt locks, the A-tool causes a lot of damage to the door itself. Instead we either use the adz of the halligan or the blade of a flat axe and drive it downwards to shear the brass bolt that holds the lock together. Placed on top of the protuding lock face, not behind it. Some of the guys are skilled enough to walk up and use the backside of the ax to swing down and strike the deadbolt, cleaning it off the door in one hit. We are currently placing K-tools in-service to replace to the less effective A-tool for the Adams style locks. I have also seen some halligans with A-tools cut/machined into the adz. We have a Chicago Patrol bar (Basically a Kelly tool with a hammer face on the back side of the adz, check out the FARRMAX tool from Firehooks Unlimited) with an A-tool machined into the adz. Many guys will carry this tool with the halligan for a more versatile set of irons".

When I'm the engine officer I carry an A tool with the TIC and a good light. The A tool will take any lock the K tool will and many more that the K tool will not. I am also a truck officer and when in that capacity I stick with the halligan and a maul as well as the A tool.
If you are trying to pry anything more then a window use something else. the A tool is limited in its uses, I would never carry it instead of something else. The reason I don't carry irons when I'm the engine officer is that I do have to put my hands on the line because of manning sometimes. So I keep the tools to a minimum.
The biggest thing with pulling locks is the door has to be appropriate for this tactic. Many appartment doors by us are cheaper wood and even though they put enough locks on them to prevent kick ins. (By bad guys or us) the door is so lightweight the K or A tool tears apart the skin of the door without pulling the cylinder. Now these doors don't stand a chance when using a good halligan, so that is what we use.